Generalized snake posets, order polytopes, and lattice-point enumeration

Building from the work of von Bell et al.~(2022), we study the Ehrhart theory of order polytopes arising from a special class of distributive lattices, known as generalized snake posets. We present arithmetic properties satisfied by the Ehrhart polynomials of order polytopes of generalized snake posets along with a computation of their Gorenstein index. Then we give a combinatorial description of the chain polynomial of generalized snake posets as a direction to obtain the $h^$-polynomial of their associated order polytopes. Additionally, we present explicit formulae for the $h^$-polynomial of the order polytopes of the two extremal examples of generalized snake posets, namely the ladder and regular snake poset. We then provide a recursive formula for the $h^$-polynomial of any generalized snake posets and show that the $h^$-vectors are entry-wise bounded by the $h^*$-vectors of the two extremal cases.

💡 Research Summary

The paper investigates the Ehrhart theory of order polytopes associated with a special family of distributive lattices called generalized snake posets. A generalized snake poset P(w) is defined by a word w over the alphabet {L,R}; starting from a four‑element base poset, each letter adds a new square face and glues it to the previous one, yielding a width‑two distributive lattice of rank n + 2 for a word of length n. Two extremal families are highlighted: the regular snake Sₙ (the alternating word LRLR…) and the ladder Lₙ (the constant word LLLL…).

The authors first recall the construction of order polytopes O(P) (Stanley, 1986) and the basic notions of Ehrhart theory: the Ehrhart polynomial L(P; t) counting lattice points in dilates, the h*‑polynomial h*(P; z) encoding the Ehrhart series, and the Gorenstein property (equivalent to the poset being graded).

Main arithmetic constraints (Theorem 2.9). For any word w of length n, the Ehrhart polynomial satisfies
1. (t + 1)(t + 2)…(t + n + 3) divides L(w; t);
2. L(w; t) = L(w; − n − 4 − t) (symmetry);
3. all integer roots lie in